# Set of vectors form a vector space

• MHB
• karush
In summary, the conversation is about addition of matrices and how it works. The given matrices have elements x, y, and 5z, which can be added together to get X, Y, and 10Z. However, in the second case, the z values are either not present or are all equal to 5, which means that the addition is not closed and cannot result in the desired output.
karush
Gold Member
MHB
View attachment 8769this is what is given
$$\begin{bmatrix}x_1\\y_1\\5z_1\end{bmatrix} \oplus \begin{bmatrix} x_2\\y_2\\5z_2 \end{bmatrix} = \begin{bmatrix} x_1+x_2\\y_1+y_2\\5z_1+5z_2 \end{bmatrix} = \begin{bmatrix} X\\Y\\10Z \end{bmatrix}$$

uhmmmm really?

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karush said:
this is what is given
$$\begin{bmatrix}x_1\\y_1\\5z_1\end{bmatrix} \oplus \begin{bmatrix} x_2\\y_2\\5z_2 \end{bmatrix} = \begin{bmatrix} x_1+x_2\\y_1+y_2\\5z_1+5z_2 \end{bmatrix} = \begin{bmatrix} X\\Y\\10Z \end{bmatrix}$$

uhmmmm really?
This time there are no z's. Or z = 5 in all cases, if you prefer to look at it that way.
$$\displaystyle \left [ \begin{matrix} x_1 \\ y_1 \\ 5 \end{matrix} \right ] \oplus \left [ \begin{matrix} x_2 \\ y_2 \\ 5 \end{matrix} \right ] = \left [ \begin{matrix} x_1 + x_2 \\ y_1 + y_2 \\ 5 + 5 \end{matrix} \right ] \notin \left [ \begin{matrix} X \\ Y \\ 5 \end{matrix} \right ]$$

so addition is not closed this time.

-Dan

https://dl.orangedox.com/GXEVNm73NxaGC9F7Cy

## 1. What is a vector space?

A vector space is a mathematical concept that describes a set of vectors that can be added together and multiplied by scalars to create new vectors. It is a fundamental concept in linear algebra and is used to model physical quantities such as velocity, force, and acceleration.

## 2. How do you know if a set of vectors forms a vector space?

A set of vectors forms a vector space if it satisfies the following properties: closure under vector addition, closure under scalar multiplication, existence of a zero vector, existence of additive inverses, and associativity and distributivity of vector operations. These properties ensure that the set is closed and consistent under vector operations.

## 3. Can a set of vectors form more than one vector space?

Yes, a set of vectors can form multiple vector spaces depending on the choice of field and the dimension of the vectors. For example, the set of all 2-dimensional vectors can form a vector space over both the real numbers and the complex numbers, but the properties of the vector space will differ depending on the field.

## 4. What is the difference between a vector space and a subspace?

A subspace is a subset of a vector space that also satisfies the properties of a vector space. In other words, a subspace is a smaller vector space contained within a larger vector space. Every vector space contains at least two subspaces: the trivial subspace consisting of only the zero vector, and the entire vector space itself.

## 5. How is a vector space related to linear independence?

Linear independence is a concept that describes whether a set of vectors can be written as a linear combination of other vectors. In a vector space, a set of linearly independent vectors can span the entire space and form a basis, while a set of linearly dependent vectors will be redundant and not add any new information to the space. Therefore, linear independence is closely related to the structure and dimension of a vector space.

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